The particle production by external time-dependent field \(V(t)\) 
was studied in many works dedicated
to the universe heating, see e.g. [1]}. In particular the production rate was calculated
perturbatively (but not only) in the book [2]} for the (inflaton) field with
the harmonic dependence on time:
\(V(\eta ) = V_0 \cos (\Omega _c \eta + \theta ),\) 
